Angermann L. (ed.). Numerical Simulations - Applications, Examples and Theory
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Generation and Resonance Scattering of Waves on
Cubically Polarisable Layered Structures
7
higher harmonics), it is possible to restrict the investigation to a system of three equations.
Taking into account only the non-trivial terms in the expansion of the polarisation coefficients,
we arrive at the following system:
⎧
⎪
⎨
⎪
⎩
∇
2
E
1
(r,κ)+ε
(L)
κ
2
E
1
(r,κ)+4πκ
2
P
(NL)
1
(r,κ)=0,
∇
2
E
1
(r,2κ)+ε
(L)
(2κ)
2
E
1
(r,2κ)+4π(2κ)
2
P
(NL)
1
(r,2κ)=0,
∇
2
E
1
(r,3κ)+ε
(L)
(3κ)
2
E
1
(r,3κ)+4π(3κ)
2
P
(NL)
1
(r,3κ)=0,
P
(NL)
1
(r,nκ)=
3
4
χ
(3)
1111
(nκ; κ,−κ, nκ)|E
1
(r,κ)|
2
+ χ
(3)
1111
(nκ;2κ,−2κ, nκ)|E
1
(r,2κ)|
2
+χ
(3)
1111
(nκ;3κ,−3κ, nκ)|E
1
(r,3κ)|
2
E
1
(r,nκ)
+
δ
n1
3
4
χ
(3)
1111
(κ;−κ, −κ,3κ)
[
E
∗
1
(r,κ)
]
2
E
1
(r,3κ)
+
χ
(3)
1111
(κ;2κ,2κ,−3κ)E
2
1
(r,2κ) E
∗
1
(r,3κ)
+δ
n2
3
4
χ
(3)
1111
(2κ;−2κ,κ,3κ)E
∗
1
(r,2κ) E
1
(r,κ)E
1
(r,3κ)
+
δ
n3
1
4
χ
(3)
1111
(3κ;κ,κ,κ)E
3
1
(r,κ)+
3
4
χ
(3)
1111
(3κ;2κ,2κ,−κ)E
2
1
(r,2κ) E
∗
1
(r,κ)
,
n
= 1, 2,3,
(20)
where δ
nm
denotes Kronecker’s symbol. Using (19), we obtain
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∇
2
E
1
(r,κ)+ε
(L)
κ
2
E
1
(r,κ)+4πκ
2
P
(FSM)
1
(r,κ)+P
(GC )
1
(r,κ)
= −4πκ
2
P
(G)
1
(r,κ),
∇
2
E
1
(r,2κ)+ε
(L)
(2κ)
2
E
1
(r,2κ)+4π(2κ)
2
P
(FSM)
1
(r,2κ)+P
(GC )
1
(r,2κ)
= 0,
∇
2
E
1
(r,3κ)+ε
(L)
(3κ)
2
E
1
r(r,3κ)+4π(3κ)
2
P
(FSM)
1
(r,3κ)
= −
4π(3κ)
2
P
(G)
1
(r,3κ),
P
(FSM)
1
(r,nκ)=
3
4
(χ
(3)
1111
(nκ; κ,−κ, nκ)|E
1
(r,κ)|
2
+ χ
(3)
1111
(nκ;2κ,−2κ, nκ)|E
1
(r,2κ)|
2
+χ
(3)
1111
(nκ;3κ,−3κ, nκ)|E
1
(r,3κ)|
2
)E
1
(r,nκ), n = 1, 2,3,
P
(GC )
1
(r,κ)=
3
4
χ
(3)
1111
(κ;−κ, −κ,3κ)
[
E
∗
1
(r,κ)
]
2
E
1
(r,3κ)
=
3
4
χ
(3)
1111
(κ;−κ, −κ,3κ)
E
∗
1
(r,κ)
2
E
1
(r,κ)
E
1
(r,3κ) E
1
(r,κ),
P
(G)
1
(r,κ)=
3
4
χ
(3)
1111
(κ;2κ,2κ,−3κ)E
2
1
(r,2κ) E
∗
1
(r,3κ),
P
(GC )
1
(r,2κ)=
3
4
χ
(3)
1111
(2κ;−2κ, κ,3κ)E
∗
1
(r,2κ) E
1
(r,κ)E
1
(r,3κ)
=
3
4
χ
(3)
1111
(2κ;−2κ, κ,3κ)
E
∗
1
(r,2κ)
E
1
(r,2κ)
E
1
(r,κ)E
1
(r,3κ) E
1
(r,2κ),
P
(G)
1
(r,3κ)=
3
4
1
3
χ
(3)
1111
(3κ;κ,κ,κ)E
3
1
(r,κ)+χ
(3)
1111
(3κ;2κ,2κ,−κ)E
2
1
(r,2κ) E
∗
1
(r,κ)
.
(21)
The analysis of the problem can be significantly simplified by reducing the number of
parameters, i.e. the coefficients of the cubic susceptibility of the non-linear medium. Thus,
181
Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures
8 Numerical Simulations, Applications, Examples and Theory
by Kleinman’s rule (Kleinman (1962), Miloslavski (2008)),
χ
(3)
1111
(nκ; κ,−κ, nκ)=χ
(3)
1111
(nκ;2κ,−2κ, nκ)=χ
(3)
1111
(nκ;3κ,−3κ, nκ)
=
χ
(3)
1111
(κ;−κ,−κ,3κ)= χ
(3)
1111
(κ;2κ,2κ,−3κ)=χ
(3)
1111
(2κ;−2κ, κ,3κ)
=
χ
(3)
1111
(3κ;κ,κ,κ)=χ
(3)
1111
(3κ;2κ,2κ,−κ)=: χ
(3)
1111
, n = 1,2,3.
Therefore, the system (21) can be written in the form
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∇
2
E
1
(r,κ)+ε
(L)
κ
2
E
1
(r,κ)+4πκ
2
P
(FSM)
1
(r,κ)+P
(GC )
1
(r,κ)
= −4πκ
2
P
(G)
1
(r,κ),
∇
2
E
1
(r,2κ)+ε
(L)
(2κ)
2
E
1
(r,2κ)+4π(2κ)
2
P
(FSM)
1
(r,2κ)+P
(GC )
1
(r,2κ)
= 0,
∇
2
E
1
(r,3κ)+ε
(L)
(3κ)
2
E
1
(r,3κ)+4π(3κ)
2
P
(FSM)
1
(r,3κ)
= −
4π(3κ)
2
P
(G)
1
(r,3κ),
P
(FSM)
1
(r,nκ)=
3
4
χ
(3)
1111
(|E
1
(r,κ)|
2
+ |E
1
(r,2κ)|
2
+ |E
1
(r,3κ)|
2
)E
1
(r,nκ), n = 1, 2,3,
P
(GC )
1
(r,κ)=
3
4
χ
(3)
1111
E
∗
1
(r,κ)
2
E
1
(r,κ)
E
1
(r,3κ) E
1
(r,κ),
P
(G)
1
(r,κ)=
3
4
χ
(3)
1111
E
2
1
(r,2κ) E
∗
1
(r,3κ),
P
(GC )
1
(r,2κ)=
3
4
χ
(3)
1111
E
∗
1
(r,2κ)
E
1
(r,2κ)
E
1
(r,κ)E
1
(r,3κ) E
1
(r,2κ), P
(G)
1
(r,2κ) := 0,
P
(G)
1
(r,3κ), =
3
4
χ
(3)
1111
1
3
E
3
1
(r,κ)+E
2
1
(r,2κ) E
∗
1
(r,κ)
, P
(GC )
1
(r,3κ) := 0.
(22)
The permittivity of the non-linear medium filling a layer (see Fig. 1) can be represented as
ε
nκ
= ε
(L)
+ ε
(NL)
nκ
for |z|≤2πδ. (23)
Outside the layer, i.e. for
|z|> 2πδ, ε
nκ
= 1. The linear and non-linear terms of the permittivity
of the layer are given by the coefficients at
(nκ)
2
E
1
(r,nκ) in the second and third addends in
each of the equations of the system, respectively. Thus
ε
(L)
=
D
(L)
1
(r,nκ)
E
1
(r,nκ)
=
1 + 4πχ
(1)
11
, (24)
where the representations for the linear part of the complex components of the electric
displacement D
(L)
1
(r,nκ)=E
1
(r,nκ)+4πP
(L)
1
(r,nκ)=ε
(L)
E
1
(r,nκ) and the polarisation
P
(L)
1
(r,nκ)=χ
(1)
11
E
1
(r,nκ) are taken into account. Similarly, the third term of each equation of
the system makes it possible to write the non-linear component of the permittivity in the form
ε
(NL)
nκ
= 4π
P
(FSM)
1
(r,nκ)+P
(GC )
1
(r,nκ)
E
1
(r,nκ)
=
α(z)
|E
1
(r,κ)|
2
+ |E
1
(r,2κ)|
2
+ |E
1
(r,3κ)|
2
+ δ
n1
E
∗
1
(r,κ)
2
E
1
(r,κ)
E
1
(r,3κ)+δ
n2
E
∗
1
(r,2κ)
E
1
(r,2κ)
E
1
(r,κ)E
1
(r,3κ)
,
(25)
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Numerical Simulations - Applications, Examples and Theory
Generation and Resonance Scattering of Waves on
Cubically Polarisable Layered Structures
9
where α(z) := 3πχ
(3)
1111
(z) is the so-called function of the cubic susceptibility of the non-linear
medium.
For transversely inhomogeneous media (a layer or a layered structure), the linear part ε
(L)
=
ε
(L)
(z)=1 + 4πχ
(1)
11
(z) of the permittivity (cf. (24)) is described by a piecewise smooth or a
piecewise constant function. Similarly, the function of the cubic susceptibility α
= α(z) is also
a piecewise smooth or a piecewise constant function. This assumption allows us to investigate
the diffraction characteristics of a non-linear layer and of a layered structure (consisting of a
finite number of non-linear dielectric layers) within one and the same mathematical model.
3. The condition of phase synchronism. Quasi-homogeneous electromagnetic
fields in a transversely inhomogeneous non-linear dielectric layered structure.
The scattered and generated field in a transversely inhomogeneous, non-linear dielectric
layer excited by a plane wave is quasi-homogeneous along the coordinate y, hence it can be
represented as
(C1) E
1
(r,nκ)=: E
1
(nκ; y,z) := U(nκ;z)exp(iφ
nκ
y), n = 1, 2,3.
Here U
(nκ; z) and φ
nκ
:= nκ sin ϕ
nκ
denote the complex-valued transverse component of the
Fourier amplitude of the electric field and the value of the longitudinal propagation constant
(longitudinal wave-number) at the frequency nκ, respectively, where ϕ
nκ
is the given angle of
incidence of the exciting field of frequency nκ (cf. Fig. 1).
The dielectric permittivities of the layered structure at the multiple frequencies nκ are
determined by the values of the transverse components of the Fourier amplitudes of the
scattered and generated fields, i.e. by the redistribution of energy of the electric fields at
multiple frequencies, where the angles of incidence are given and the non-linear structure
under consideration is transversely inhomogeneous. The condition of the longitudinal
homogeneity (along the coordinate y) of the non-linear dielectric constant of the layered
structure can be written as
ε
(NL)
nκ
(z,α(z), E
1
(r,κ), E
1
(r,2κ), E
1
(r,3κ)) = ε
(NL)
nκ
(z,α(z),U(κ; z),U(2κ;z),U(3κ; z)). (26)
Using the representation (25) and the conditions (C1), (26), we obtain the following physically
consistent requirement, which we call the condition of the phase synchronism of waves:
(C2) φ
nκ
= nφ
κ
, n = 1,2,3.
Indeed, from (25) and (C1) it follows that
ε
(NL)
nκ
= α(z)
|E
1
(r,κ)|
2
+ |E
1
(r,2κ)|
2
+ |E
1
(r,3κ)|
2
+ δ
n1
E
∗
1
(r,κ)
2
E
1
(r,κ)
E
1
(r,3κ)+δ
n2
E
∗
1
(r,2κ)
E
1
(r,2κ)
E
1
(r,κ)E
1
(r,3κ)
,
= α(z)
|U(κ; z)|
2
+ |U(2κ; z)|
2
+ |U(3κ; z)|
2
+ δ
n1
[
U
∗
(r,κ)
]
2
U(r,κ)
U(3κ; z) exp
{
i
[
−
3φ
κ
+ φ
3κ
]
y
}
+
δ
n2
U
∗
(r,2κ)
U(r,2κ)
U(κ; z)U(3κ;z)exp
{
i
[
−
2φ
2κ
+ φ
κ
+ φ
3κ
]
y
}
, n
= 1, 2,3.
(27)
Therefore the condition (26) is satisfied if
−3φ
κ
+ φ
3κ
= 0,
−2φ
2κ
+ φ
κ
+ φ
3κ
= 0.
(28)
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Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures
10 Numerical Simulations, Applications, Examples and Theory
From this system we obtain the condition (C2).
According to (23), (24), (27) and (C2), the permittivity of the non-linear layer can be expressed
as
ε
nκ
(z,α(z), E
1
(r,κ), E
1
(r,2κ), E
1
(r,3κ))
=
ε
nκ
(z,α(z),U(κ; z),U(2κ;z),U(3κ; z))
=
ε
(L)
(z)+α(z)
|U(κ; z)|
2
+ |U(2κ; z)|
2
+ |U(3κ; z)|
2
+ δ
n1
U
∗
(κ;z)exp
{
−
2iarg( U(κ;z))
}
U(3κ; z)
+
δ
n2
exp
{
−
2iarg( U(2κ;z))
}
U(κ, z)U(3κ;z)
= ε
(L)
(z)+α(z)
|U( κ; z)|
2
+ |U(2κ; z)|
2
+ |U(3κ; z)|
2
+ δ
n1
|U( κ; z)||U(3κ;z)|exp
{
i
[
−
3arg(U(κ; z)) −3φ
κ
y + arg(U(3κ; z)) + φ
3κ
y
]
}
+
δ
n2
|U( κ; z)||U(3κ;z)|exp
{
i
[
−
2arg(U(2κ; z)) −2φ
2κ
y + arg(U(κ; z)) + φ
κ
y
+ arg(U(3κ; z)) + φ
3κ
y
]
}
= ε
(L)
(z)+α(z)
|U(κ; z)|
2
+ |U(2κ; z)|
2
+ |U(3κ; z)|
2
+ δ
n1
|U(κ; z)||U(3κ;z)|exp
{
i
[
−
3arg(U(κ; z)) + arg(U(3κ; z))
]
}
+
δ
n2
|U(κ; z)||U(3κ;z)|exp
{
i
[
−
2arg(U(2κ; z)) + arg(U(κ; z)) + arg(U(3κ; z))
]
}
,
n
= 1, 2,3.
(29)
The investigation of the quasi-homogeneous fields E
1
(nκ; y,z) (cf. condition (C1)) in a
transversely inhomogeneous non-linear dielectric layer shows that, if the condition of the
phase synchronism (C2) is satisfied, the components of the non-linear polarisation P
(G)
1
(r,nκ)
(playing the role of the sources generating radiation in the right-hand sides of the system (22))
satisfy the quasi-homogeneity condition, too. Indeed, using (25) and (C1), the right-hand sides
of the first and third equations of (22) can be rewritten as
−4πκ
2
P
(G)
1
(r,κ)=−α(z)κ
2
E
2
1
(r,2κ) E
∗
1
(r,3κ)
= −
α(z)κ
2
U
2
(2κ;z)U
∗
(3κ;z)exp
{
i
[
2φ
2κ
−φ
3κ
]
y
}
= −
α(z)κ
2
U
2
(2κ;z)U
∗
(3κ;z)exp(iφ
κ
y)
and
−4π(3κ)
2
P
(G)
1
(r,3κ)=−α(z)(3κ)
2
1
3
E
3
1
(r,κ)+E
2
1
(r,2κ) E
∗
1
(r,κ)
= −α(z)(3κ)
2
1
3
U
3
(κ;z)exp(3iφ
κ
y)
+
U
2
(2κ;z)U
∗
(κ;z)exp
{
i
[
2φ
2κ
−φ
κ
]
y
}
= −α(z)(3κ)
2
1
3
U
3
(κ;z)+U
2
(2κ;z)U
∗
(κ;z)
exp
(iφ
3κ
y),
respectively. This shows that the quasi-homogeneity condition for the components of the
non-linear polarisation P
(G)
1
(r,nκ) is satisfied.
In the considered case of spatially quasi-homogeneous (along the coordinate y)
electromagnetic fields (C1), the condition of the phase synchronism of waves (C2) reads as
sin ϕ
nκ
= sin ϕ
κ
, n = 1,2,3.
Consequently, the given angle of incidence of a plane wave at the frequency κ coincides with
the possible directions of the angles of incidence of plane waves at the multiple frequencies
nκ. The angles of the wave scattered by the layer are equal to ϕ
scat
nκ
= −ϕ
nκ
in the zone of
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Numerical Simulations - Applications, Examples and Theory
Generation and Resonance Scattering of Waves on
Cubically Polarisable Layered Structures
11
reflection z > 2πδ and ϕ
scat
nκ
= π + ϕ
nκ
and in the zone of transmission of the non-linear layer
z
< −2πδ, where all angles are measured counter-clockwise in the (y,z)-plane from the z-axis
(cf. Fig. 1).
4. The diffraction of a packet of plane waves on a non-linear layered dielectric
structure. The third harmonics generation
As a first observation we mention that the effect of a weak quasi-homogeneous
electromagnetic field (C1) on the non-linear dielectric structure such that harmonics at
multiple frequencies are not generated, i.e. E
1
(r,2κ)=0 and E
1
(r,3κ)=0, reduces to find the
electric field component E
1
(r,κ) determined by the first equation of the system (22). In this
case, a diffraction problem for a plane wave on a non-linear dielectric layer with a Kerr-type
non-linearity ε
nκ
= ε
(L)
(z)+α(z)|E
1
(r,κ)|
2
and a vanishing right-hand side is to be solved,
see Yatsyk (2007); Shestopalov & Yatsyk (2007); Kravchenko & Yatsyk (2007); Angermann &
Yatsyk (2008); Yatsyk (2006); Smirnov et al. (2005); Serov et al. (2004).
The generation process of a field at the triple frequency 3κ by the non-linear dielectric structure
is caused by a strong incident electromagnetic field at the frequency κ and can be described
by the first and third equations of the system (22) only. Since the right-hand side of the second
equation in (22) is equal to zero, we may set E
1
(r,2κ)=0 corresponding to the homogeneous
boundary condition w.r.t. E
1
(r,2κ). Therefore the second equation in (22) can be completely
omitted.
A further interesting problem consists in the investigation of the influence of a packet of waves
on the generation of the third harmonic, if a strong incident field at the basic frequency κ and,
in addition, weak incident quasi-homogeneous electromagnetic fields at the double and triple
frequencies 2κ,3κ (which alone do not generate harmonics at multiple frequencies) excite the
non-linear structure. The system (22) allows to describe the corresponding process of the third
harmonics generation. Namely, if such a wave packet consists of a strong field at the basic
frequency κ and of a weak field at the triple frequency 3κ, then we arrive, as in the situation
described above, at the system (22) with E
1
(r,2κ)=0, i.e. it is sufficient to consider the first
and third equations of (22) only. For wave packets consisting of a strong field at the basic
frequency κ and of a weak field at the frequency 2κ, (or of two weak fields at the frequencies
2κ and 3κ) we have to take into account all three equations of system (22). This is caused by
the inhomogeneity of the corresponding diffraction problem, where a weak incident field at
the double frequency 2κ (or two weak fields at the frequencies 2κ and 3κ) excites (resp. excite)
the dielectric medium.
So we consider the problem of diffraction of a packet of plane waves consisting of a strong
field at the frequency κ (which generates a field at the triple frequency 3κ) and of weak fields
at the frequencies 2κ and 3κ (having an impact on the process of third harmonic generation
due to the contribution of weak electromagnetic fields of diffraction)
E
inc
1
(r,κ) := E
inc
1
(κ;y,z) := a
inc
nκ
exp
i
φ
nκ
y − Γ
nκ
(z −2πδ)
3
n
=1
, z > 2πδ, (30)
with amplitudes a
inc
nκ
and angles of incidence ϕ
nκ
, |ϕ| < π/2 (cf. Fig. 1), where φ
nκ
:=
nκ sin ϕ
nκ
are the longitudinal propagation constants (longitudinal wave-numbers) and Γ
nκ
:=
(
nκ
)
2
−φ
2
nκ
are the transverse propagation constants (transverse wave-numbers).
In this setting, the complex amplitudes of the total fields of diffraction
E
1
(r,nκ)=: E
1
(nκ; y,z) := U(nκ;z)exp(iφ
nκ
y) := E
inc
1
(nκ; y,z)+E
scat
1
(nκ; y,z)
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Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures
12 Numerical Simulations, Applications, Examples and Theory
of a plane wave (30) in a non-magnetic, isotropic, linearly polarised
E
(r,nκ)=
(
E
1
(nκ; y,z),0,0
)
,
H
(r,nκ)=
0,
1
inωμ
0
∂E
1
(nκ; y,z)
∂z
,
−
1
inωμ
0
∂E
1
(nκ; y,z)
∂y
(E-polarisation), transversely inhomogeneous ε
(L)
= ε
(L)
(z)=1 + 4πχ
(1)
11
(z) dielectric layer
(see Fig. 1) with a cubic polarisability P
(NL)
(r,nκ)=(P
(NL)
1
(nκ; y,z),0,0)
of the medium (see
(20)) satisfies the system of equations (cf. (22) – (25))
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∇
2
E
1
(r,κ)+κ
2
ε
κ
(z,α(z), E
1
(r,κ), E
1
(r,2κ), E
1
(r,3κ))E
1
(r,κ)
= −
α(z)κ
2
E
2
1
(r,2κ) E
∗
1
(r,3κ),
∇
2
E
1
(r,2κ)+(2κ)
2
ε
2κ
(z,α(z), E
1
(r,κ), E
1
(r,2κ), E
1
(r,3κ))E
1
(r,2κ)=0,
∇
2
E
1
(r,3κ)+(3κ)
2
ε
3κ
(z,α(z), E
1
(r,κ), E
1
(r,2κ), E
1
(r,3κ))E
1
(r,3κ)
= −
α(z)(3κ)
2
1
3
E
3
1
(r,κ)+ E
2
1
(r,2κ) E
∗
1
(r,κ)
(31)
together with the following conditions, where E
tg
(nκ; y,z) and H
tg
(
nκ;y,z
)
denote
the tangential components of the intensity vectors of the full electromagnetic field
{
E(nκ;y,z)
}
n=1,2,3
,
{
H(nκ;y,z)
}
n=1,2,3
:
– (C1) E
1
(nκ; y,z)=U(nκ;z)exp(iφ
nκ
y), n = 1, 2,3
(the quasi-homogeneity condition w.r.t. the spatial variable y introduced in Section 3),
– (C2) φ
nκ
= nφ
κ
, n = 1,2,3,
(the condition of phase synchronism of waves introduced in Section 3),
– (C3) E
tg
(nκ; y,z) and H
tg
(nκ; y,z) (i.e. E
1
(nκ; y,z) and H
2
(nκ; y,z)) are continuous at the
boundary layers of the non-linear structure,
– (C4) E
scat
1
(nκ; y,z)=
a
scat
nκ
b
scat
nκ
exp
(
i
(
φ
nκ
y ± Γ
nκ
(z ∓2πδ)
))
, z
>
<
±
2πδ, n = 1, 2,3
(the radiation condition w.r.t. the scattered field).
The condition (C4) provides a physically consistent behaviour of the energy characteristics
of scattering and guarantees the absence of waves coming from infinity (i.e. z
= ±∞), see
Shestopalov & Sirenko (1989). We study the scattering properties of the non-linear layer,
where in (C4) we always have Im Γ
nκ
= 0, Re Γ
nκ
> 0. Note that (C4) is also applicable for
the analysis of the wave-guide properties of the layer, where Im Γ
nκ
> 0, Re Γ
nκ
= 0.
Here and in what follows we use the following notation:
(r,t) are dimensionless
spatial-temporal coordinates such that the thickness of the layer is equal to 4πδ. The
time-dependence is determined by the factors exp
(−inωt), where ω := κc is the dimensionless
circular frequency and κ is a dimensionless frequency parameter such that κ
= ω/c := 2π/λ.
This parameter characterises the ratio of the true thickness h of the layer to the free-space
wavelength λ, i.e. h/λ
= 2κδ. c =(ε
0
μ
0
)
−1/2
denotes a dimensionless parameter, equal to the
absolute value of the speed of light in the medium containing the layer (Im c
= 0). ε
0
and μ
0
are the material parameters of the medium. The absolute values of the true variables r
,t
,ω
are given by the formulas r
= hr/4πδ, t
= th/4πδ, ω
= ω4πδ/h.
186
Numerical Simulations - Applications, Examples and Theory
Generation and Resonance Scattering of Waves on
Cubically Polarisable Layered Structures
13
The desired solution of the diffraction problem (31), (C1) – (C4) can be represented as follows:
E
1
(nκ; y,z)=U(nκ;z)exp(iφ
nκ
y)
=
⎧
⎨
⎩
a
inc
nκ
exp(i(φ
nκ
y − Γ
nκ
(z −2πδ))) + a
scat
nκ
exp(i(φ
nκ
y + Γ
nκ
(z −2πδ))), z > 2πδ,
U
(nκ; z)exp(iφ
nκ
y), |z|≤2πδ,
b
scat
nκ
exp(i(φ
nκ
y − Γ
nκ
(z + 2πδ))), z < −2πδ,
n
= 1, 2,3.
(32)
Note that depending on the magnitudes of the amplitudes
{a
inc
κ
, a
inc
2κ
, a
inc
3κ
} of the packet of
incident plane waves, the amplitudes
{a
scat
nκ
,b
scat
nκ
}
3
n
=1
of the scattered fields can be considered
as the amplitudes of the diffraction field, of the generation field or of the sum of the diffraction
and generation fields. If the components
{a
inc
κ
= a
inc(w)
κ
, a
inc
2κ
= a
inc(w)
2κ
, a
inc
3κ
= a
inc(w)
3κ
} of the
packet consist of the amplitudes of weak fields, then
{a
scat
nκ
= a
dif
nκ
,b
scat
nκ
= b
dif
nκ
}
3
n
=1
.
The presence of an amplitude of a strong field at the basic frequency κ in the packet
{a
inc
κ
=
a
inc(s)
κ
, a
inc
2κ
= a
inc(w)
2κ
, a
inc
3κ
= a
inc(w)
3κ
} leads to non-trivial right-hand sides in the problem (31),
(C1) – (C4). In this case the analysis of the following situations is of interest (see (32)):
⎧
⎪
⎨
⎪
⎩
a
inc
κ
= a
inc(s)
κ
= 0,
a
inc
2κ
= a
inc(w)
2κ
:= 0,
a
inc
3κ
= a
inc(w)
3κ
:= 0
⎫
⎪
⎬
⎪
⎭
⇒
a
scat
κ
= a
dif
κ
, a
scat
2κ
= 0, a
scat
3κ
= a
gen
3κ
b
scat
κ
= b
dif
κ
, b
scat
2κ
= 0, b
scat
3κ
= b
gen
3κ
,
⎧
⎪
⎨
⎪
⎩
a
inc
κ
= a
inc(s)
κ
= 0,
a
inc
2κ
= a
inc(w)
2κ
:= 0,
a
inc
3κ
= a
inc(w)
3κ
= 0
⎫
⎪
⎬
⎪
⎭
⇒
a
scat
κ
= a
dif
κ
, a
scat
2κ
= 0, a
scat
3κ
= a
dif
3κ
+ a
gen
3κ
b
scat
κ
= b
dif
κ
, b
scat
2κ
= 0, b
scat
3κ
= b
dif
3κ
+ b
gen
3κ
,
⎧
⎪
⎨
⎪
⎩
a
inc
κ
= a
inc(s)
κ
= 0,
a
inc
2κ
= a
inc(w)
2κ
= 0,
a
inc
3κ
= a
inc(w)
3κ
:= 0
⎫
⎪
⎬
⎪
⎭
⇒
a
scat
κ
= a
dif
κ
+ a
gen
κ
, a
scat
2κ
= a
dif
2κ
, a
scat
3κ
= a
gen
3κ
b
scat
κ
= b
dif
κ
+ b
gen
κ
, b
scat
2κ
= b
dif
2κ
, b
scat
3κ
= b
gen
3κ
,
⎧
⎪
⎨
⎪
⎩
a
inc
κ
= a
inc(s)
κ
= 0,
a
inc
2κ
= a
inc(w)
2κ
= 0,
a
inc
3κ
= a
inc(w)
3κ
= 0
⎫
⎪
⎬
⎪
⎭
⇒
a
scat
κ
= a
dif
κ
+ a
gen
κ
, a
scat
2κ
= a
dif
2κ
, a
scat
3κ
= a
dif
3κ
+ a
gen
3κ
b
scat
κ
= b
dif
κ
+ b
gen
κ
, b
scat
2κ
= b
dif
2κ
, b
scat
3κ
= b
dif
3κ
+ b
gen
3κ
.
The boundary conditions follow from the continuity of the tangential components of the full
fields of diffraction
E
tg
(nκ; y,z)
n=1,2,3
H
tg
(nκ; y,z)
n=1,2,3
at the boundary z = 2πδ and
z
= −2πδ of the non-linear layer (cf. (C3)). According to (C3) and the presentation of the
electrical components of the electromagnetic field (32), at the boundary of the non-linear layer
we obtain:
U
(nκ;2πδ)=a
scat
nκ
+ a
inc
nκ
, U
(nκ;2πδ)=iΓ
nκ
a
scat
nκ
− a
inc
nκ
,
U
(nκ; −2πδ)=b
scat
nκ
, U
(nκ; −2πδ)=−iΓ
nκ
b
scat
nκ
, n = 1,2,3,
(33)
where “
” denotes the differentiation w.r.t. z. Eliminating in (33) the unknown values of
the complex amplitudes
a
scat
nκ
n=1,2,3
,
b
scat
nκ
n=1,2,3
of the scattered field and taking into
consideration that a
inc
nκ
= U
inc
(nκ;2πδ), we arrive at the desired boundary conditions for the
problem (31), (C1) – (C4):
iΓ
nκ
U(nκ;−2πδ)+U
(nκ; −2πδ)=0,
iΓ
nκ
U(nκ;2πδ) − U
(nκ;2πδ)=2iΓ
nκ
a
inc
nκ
, n = 1,2,3.
(34)
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Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures
14 Numerical Simulations, Applications, Examples and Theory
Substituting the representation (32) for the desired solution into the system (31), the
resulting system of non-linear ordinary differential equations together with the boundary
conditions (34) forms a semi-linear boundary-value problem of Sturm-Liouville type, see
also Shestopalov & Yatsyk (2010); Yatsyk (2007); Shestopalov & Yatsyk (2007); Angermann
& Yatsyk (2010).
5. The system of non-linear integral equations
Similarly to the results given in Yatsyk (2007); Shestopalov & Yatsyk (2007); Kravchenko &
Yatsyk (2007); Angermann & Yatsyk (2010); Shestopalov & Sirenko (1989), the problem (31),
(C1) – (C4) reduces to finding solutions of one-dimensional non-linear integral equations
(along the height z
∈
(
−2πδ,2πδ
)
of the structure) w.r.t. the components U(nκ; z), n = 1, 2,3, of
the fields scattered and generated in the non-linear layer. We give the derivation of this system
of equations in the case of excitation of the non-linear structure by a plane wave packet (30).
The solution of (31), (C1) – (C4) in the whole space Q :
=
{
q =(y, z) : |y| < ∞, |z| < ∞
}
is
obtained using the properties of the canonical Green’s function of the problem (31), (C1)
– (C4) (for the special case ε
nκ
≡ 1) which is defined, for Y > 0, in the strip Q
{Y,∞}
:=
{
q =(y, z) : |y| < Y, |z| < ∞
}
⊂
Q by
G
0
(nκ; q,q
0
)
:=
i
4Y
exp
{
i
[
φ
nκ
(
y − y
0
)
+
Γ
nκ
|z −z
0
|
]
}
/Γ
nκ
= exp
(
±
iφ
nκ
y
)
iπ
4Y
∞
−∞
H
(1)
0
nκ
(
˘
y
−y
0
)
2
+(z − z
0
)
2
exp
(
∓
iφ
nκ
˘
y
)
d
˘
y,
n
= 1, 2,3
(35)
(cf. Shestopalov & Sirenko (1989); Sirenko et al. (1985)).
We derive the system of non-linear integral equations by the same classical approach as
described in Smirnov (1981) (see also Shestopalov & Yatsyk (2007)). Denote both the scattered
and the generated full fields of diffraction at each frequency nκ, n
= 1,2,3, i.e. the solution of
the problem (31), (C1) – (C4), by E
1
nκ;q
|
q=(y ,z)
= U(nκ;z)exp
(
iφ
nκ
y
)
(cf. (32)), and write
the system of equations (31) in the form
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∇
2
+ κ
2
E
1
(κ;q)=
[
1 −ε
κ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)]
κ
2
E
1
(κ;q)
−
α(q)κ
2
E
2
1
(2κ;q)E
∗
1
(3κ;q),
∇
2
+
(
2κ
)
2
E
1
(2κ;q)=
[
1 −ε
2κ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)](
2κ
)
2
E
1
(2κ;q),
∇
2
+(3κ)
2
E
1
(3κ;q)=
[
1 −ε
3κ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)]
(3κ)
2
E
1
(3κ;q)
−
α(q)(3κ)
2
1
3
E
3
1
(κ;q)+E
2
1
(2κ;q)E
∗
1
(κ;q)
,
or, shorter,
∇
2
+(nκ)
2
E
1
(nκ; q)=
[
1 −ε
nκ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)]
(nκ)
2
E
1
(nκ; q)
− δ
n1
α(q)(nκ)
2
E
2
1
(2κ;q)E
∗
1
(3κ;q)
−
δ
n3
α(q)(nκ)
2
1
3
E
3
1
(κ;q)+E
2
1
(2κ;q)E
∗
1
(κ;q)
, n = 1,2,3.
(36)
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Numerical Simulations - Applications, Examples and Theory
Generation and Resonance Scattering of Waves on
Cubically Polarisable Layered Structures
15
At the right-hand side of the system of equations (36), the first term outside the layer vanishes,
since, by assumption, the permittivity of the medium in which the non-linear layer is situated
is equal to one, i.e. 1
−ε
nκ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)
≡ 0 for |z| > 2πδ.
The excitation field of the non-linear structure can be represented in the form of a packet
of incident plane waves
E
inc
1
(nκ; q)
n=1,2,3
satisfying the condition of phase synchronism,
where
E
inc
1
(nκ; q)=a
inc
nκ
exp
{
i
[
φ
nκ
y − Γ
nκ
(z −2πδ)
]
}
, n = 1,2,3. (37)
Furthermore, in the present situation described by the system of equations (36), we assume
that the excitation field E
inc
1
(κ;q) of the non-linear structure at the frequency κ is sufficiently
strong (i.e. the amplitude a
inc
κ
is sufficiently large such that the third harmonic generation
is possible), whereas the amplitudes a
inc
2κ
, a
inc
3κ
corresponding to excitation fields E
inc
1
(2κ;q),
E
inc
1
(3κ;q) at the frequencies 2κ,3κ, respectively, are selected sufficiently weak such that no
generation of multiple harmonics occurs.
In the whole space, for each frequency nκ, n
= 1, 2,3, the fields
E
inc
1
(nκ; q)
n=1,2,3
of incident
plane waves satisfy a system of homogeneous Helmholtz equations:
∇
2
+(nκ)
2
E
inc
1
(nκ; q)=0, q ∈ Q, n = 1,2, 3. (38)
For z
> 2πδ, the incident fields
E
inc
1
(nκ; q)
n=1,2,3
are fields of plane waves approaching the
layer, while, for z
< 2πδ, they move away from the layer and satisfy the radiation condition
(since, in the representation of the fields E
inc
1
(nκ; q), n = 1,2,3, the transverse propagation
constants Γ
nκ
> 0, n = 1, 2,3 are positive).
Subtracting the incident fields E
inc
1
(nκ; q), from the corresponding total fields E
1
(nκ; q), cf.
(32), we obtain the following equations w.r.t. the scattered fields E
1
(nκ; q) − E
inc
1
(nκ; q)=:
E
scat
1
(nκ; q) in the zone of reflection z > 2πδ, the fields E
1
(nκ; q), |z|≤2πδ, scattered in the
layer and the fields E
1
(nκ; q)=: E
scat
1
(nκ; q), z < 2πδ, passing through the layer:
∇
2
+(nκ)
2
[
E
1
(nκ; q) − E
inc
1
(nκ; q)
= 0, z > 2πδ,
∇
2
+(nκ)
2
E
1
(nκ; q)=
[
1 −ε
nκ
(
q, α(q), E
1
(κ;q), E
1
(2κ;q), E
1
(3κ;q)
)]
(nκ)
2
E
1
(nκ; q)
−
δ
n1
α(q)(nκ)
2
E
2
1
(2κ;q)E
∗
1
(3κ;q)
−
δ
n3
α(q)(nκ)
2
1
3
E
3
1
(κ;q)+E
2
1
(2κ;q)E
∗
1
(κ;q)
,
|z|≤2πδ,
∇
2
+(nκ)
2
E
1
(nκ; q)=0, z < −2πδ, n = 1,2, 3.
(39)
Since the canonical Green’s functions satisfy the equations
∇
2
+(nκ)
2
G
0
(nκ; q,q
0
)=−δ(q,q
0
), n = 1, 2, 3, (40)
where δ
(q,q
0
) denotes the Dirac delta-function, it is easy to obtain from the above equations
(39), with q replaced by q
0
, the following system:
189
Generation and Resonance Scattering of Waves on Cubically Polarisable Layered Structures
16 Numerical Simulations, Applications, Examples and Theory
E
1
(nκ; q
0
) − E
inc
1
(nκ; q
0
)
∇
2
G
0
(nκ; q,q
0
) − G
0
(nκ; q,q
0
)∇
2
E
1
(nκ; q
0
) − E
inc
1
(nκ; q
0
)
= −
E
1
(nκ; q
0
) − E
inc
1
(nκ; q
0
)
δ
(q,q
0
), z > 2πδ,
E
1
(nκ; q
0
)∇
2
G
0
(nκ; q,q
0
) − G
0
(nκ; q,q
0
)∇
2
E
1
(nκ; q
0
)
= −
E
1
(nκ; q
0
)δ(q,q
0
)
−
G
0
(nκ; q,q
0
)
[
1 −ε
nκ
(
q
0
,α(q
0
), E
1
(κ;q
0
), E
1
(2κ;q
0
), E
1
(3κ;q
0
)
)]
(nκ)
2
E
1
(nκ; q
0
)
+
δ
n1
G
0
(nκ; q,q
0
)α(q)(nκ)
2
E
2
1
(2κ;q)E
∗
1
(3κ;q)
+
δ
n3
G
0
(nκ; q,q
0
)α(q)(nκ)
2
1
3
E
3
1
(κ;q)+E
2
1
(2κ;q)E
∗
1
(κ;q)
,
|z|≤2πδ,
E
1
(nκ; q
0
)∇
2
G
0
(nκ; q,q
0
) − G
0
(nκ; q,q
0
)∇
2
E
1
(nκ; q
0
)
= −
E
1
(nκ; q
0
)δ(q,q
0
), z < −2πδ, n = 1,2, 3.
(41)
Given Y
> 0, Z > 2πδ, now we consider in the space Q the rectangular domain
Q
{Y,Z}
:=
{
q =(y, z) : |y| < Y, |z| < Z
}
,
and the subsets
Q
{Y,Z}, z>2πδ
:=
{
q =(y, z) : |y| < Y,2πδ < z ≤ Z
}
,
Q
{Y,Z}, |z|≤2πδ
:=
{
q =(y, z) : |y| < Y, |z|≤2πδ
}
,
Q
{Y,Z}, z<−2πδ
:=
{
q =(y, z) : |y| < Y, −Z ≤ z < −2πδ
}
,
and make use of Green’s formula.
We also mention that in the case of a non-linear layered structure consisting of a finite
number of layers the applicability of Green’s formula in the region Q
{Y,Z}, |z|≤2πδ
occupied
by the dielectric follows from the continuity condition (C3) w.r.t. E
tg
(nκ; q), H
tg
(nκ; q) at
the boundaries. Indeed, consider a covering of Q
{Y,Z}
by a finite number of disjoint
rectangles such that the restrictions of ε
nκ
(
q
0
,α(q
0
), E
1
(κ;q
0
), E
1
(2κ;q
0
), E
1
(3κ;q
0
)
)
to each
of these rectangles are smooth functions. At the common interfaces of these regions (i.e.
at the boundaries of the separate layers of the structure) due to the continuity of the
components E
tg
(nκ; q) and H
tg
(nκ; q) of the electromagnetic field (cf. (C3)), E
1
(nκ; q) and
∂E
1
(nκ; q)/∂n are continuous (where n denotes the outward unit normal w.r.t. each of the
regions). Now, by Green’s formula and condition (C3) it is easy to obtain the system of
non-linear integral equations w.r.t. the unknown solutions E
1
(nκ; q), n = 1,2, 3, in the region
Q
{Y,Z}, |z|≤2πδ
. This system forms an integral representation of the solution in the exterior
Q
{Y,Z}
\ Q
{Y,Z}, |z|≤2πδ
of the region occupied by the dielectric layer. Consequently, the
desired functions
{
E
1
(nκ; q)
}
n=1,2,3
, which are twice continuously differentiable both within
(i.e. Q
{Y,Z}, |z|≤2πδ
) and outside (i.e. Q
{Y,Z}, |z|>2πδ
) of the region occupied by the dielectric
layer, are continuous and have continuous derivatives throughout the whole region Q
{Y,Z}
up
to and including the boundary ∂Q
{Y,Z}
, i.e. E
1
(nκ; q) ∈C
2
Q
{Y,Z}
∩C
1
Q
{Y,Z}
, n
= 1,2,3.
The system of non-linear integral equations and the corresponding integral representations
of the desired solution are obtained by applying, in each of the rectangles Q
{Y,Z}, z>2πδ
,
Q
{Y,Z}, |z|≤2πδ
, Q
{Y,Z}, z<−2πδ
, Green’s formula to the functions E
1
(nκ; q
0
) − E
inc
1
(
nκ;q
0
)
=
:
E
scat
1
(nκ; q
0
) for q
0
∈ Q
{Y,Z}, z>2πδ
, E
1
(nκ; q
0
)=: E
scat
1
(nκ; q
0
) for q
0
∈ Q
{Y,Z}, |z|≤2πδ
,
E
1
(nκ; q
0
)=: E
scat
1
(nκ; q
0
) for q
0
∈ Q
{Y,Z}
, z < −2πδ, and G
0
(nκ; q,q
0
) for q, q
0
∈ Q
{Y,Z}
:
190
Numerical Simulations - Applications, Examples and Theory